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| {{no footnotes|date=March 2013}}
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| In [[mathematics]] an '''automorphic number''' (sometimes referred to as a '''circular number''') is a [[number]] whose [[square number|square]] "ends" in the same digits as number itself. For example, 5<sup>2</sup> = 2'''5''', 6<sup>2</sup> = 3'''6''', 76<sup>2</sup> = 57'''76''', and 890625<sup>2</sup> = 793212'''890625''', so 5, 6, 76 and 890625 are all automorphic numbers.
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| The sequence of automorphic numbers begins [[1 (number)|1]], [[5 (number)|5]], [[6 (number)|6]], [[25 (number)|25]], [[76 (number)|76]], [[376 (number)|376]], [[625 (number)|625]], 9376, ... {{OEIS|id=A003226}}.
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| Given a ''k''-digit automorphic number <math>n>1</math>, an automorphic number <math>n'</math> with at most 2''k''-digits can be found by from the formula:
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| :<math>n'=(3\cdot n^2 - 2\cdot n^3) \ \bmod{10^{2k}}\, .</math>
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| For ''k'' greater than 1, there are at most two automorphic numbers with ''k'' [[numerical digit|digit]]s, one ending in 5 and one ending in 6. One of them has the form:
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| :<math>n\equiv 0 \pmod{2^{k}}, \quad n\equiv 1 \pmod{5^{k}}\, ,</math> | |
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| and the other has the form:
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| :<math>n\equiv 1 \pmod{2^{k}}, \quad n\equiv 0 \pmod{5^{k}}\, .</math> | |
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| The sum of the two numbers is 10<sup>k</sup> + 1. The smaller of these two numbers may be less than 10<sup>k-1</sup>; for example with ''k'' = 4 the two numbers are 9376 and 625. In this case there is only one ''k'' digit automorphic number; the smaller number could only form a ''k'' digit automorphic number if a leading 0 were added to its digits.
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| The following digit sequence can be used to find the two ''k''-digit automorphic numbers, where <math>k\leq1000</math>.
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| 12781 25400 13369 00860 34889 08436 40238 75765 93682 19796
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| 26181 91783 35204 92704 19932 48752 37825 86714 82789 05344
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| 89744 01426 12317 03569 95484 19499 44461 06081 46207 25403
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| 65599 98271 58835 60350 49327 79554 07419 61849 28095 20937
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| 53026 85239 09375 62839 14857 16123 67351 97060 92242 42398
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| 77700 75749 55787 27155 97674 13458 99753 76955 15862 71888
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| 79415 16307 56966 88163 52155 04889 82717 04378 50802 84340
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| 84412 64412 68218 48514 15772 99160 34497 01789 23357 96684
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| 99144 73895 66001 93254 58276 78000 61832 98544 26232 82725
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| 75561 10733 16069 70158 64984 22229 12554 85729 87933 71478
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| 66323 17240 55157 56102 35254 39949 99345 60808 38011 90741
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| 53006 00560 55744 81870 96927 85099 77591 80500 75416 42852
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| 77081 62011 35024 68060 58163 27617 16767 65260 93752 80568
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| 44214 48619 39604 99834 47280 67219 06670 41724 00942 34466
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| 19781 24266 90787 53594 46166 98508 06463 61371 66384 04902
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| 92193 41881 90958 16595 24477 86184 61409 12878 29843 84317
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| 03248 17342 88865 72737 66314 65191 04988 02944 79608 14673
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| 76050 39571 96893 71467 18013 75619 05546 29968 14764 26390
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| 39530 07319 10816 98029 38509 89006 21665 09580 86381 10005
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| 57423 42323 08961 09004 10661 99773 92256 25991 82128 90625 {{OEIS|id=A018247}}
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| One automorphic number is found by taking the last ''k'' digits of this sequence; the second is found by subtracting the first number from <math>10^k + 1</math>.
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| ==Other radixes== | |
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| Automorphic numbers are radix dependent, and the description above applies to automorphic numbers in [[base 10]]. Using other radixes there are different automorphic numbers. 0 and 1 are automorphic numbers in any radix; automorphic numbers other than 0 and 1 only exist when the radix has at least two distinct prime factors.
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| A single digit number ''x'' is automorphic in radix ''b'' > ''x'' when ''b'' divides ''x''<sup>2</sup> − ''x''. So 6 is automorphic in a radix which is a divisor of 6<sup>2</sup> − 6 = 30 that is greater than 6; these divisors are 10, 15 and 30.
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| In any given radix there are 2<sup>''p''</sup> sequences of automorphic numbers where ''p'' is the number of distinct prime factors in the radix. For base 10 this gives 2<sup>2</sup> = 4 sequences, which are 0,1,5 and 6 for 1 digit or 00, 01, 25, 76 for two digits and so on. A prime radix (such as 2,3,4,5,7,8,9,11,13,16,17,...) can only have 0 and 1 (prepended by one or more zeroes) as automorphic numbers. Base 6 is the first radix with non-trivial automorphic numbers and base 15 the first such odd radix. Base 30 is the first radix with three distinct prime factors and has 8 sequences of automorphic numbers. Here some examples of non-trivial 1,2 and 4 digit automorphic numbers in other radixes (using A-Z except I and O to represent digits 10 to 34):
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| {| width=100%
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| |- bgcolor="EFEFEF"
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| ! ''number of digits (n)''
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| ! ''radix''
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| ! ''numbers formed by last n digits''
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| ! ''expressed in decimal''
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| |-----
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| | 1 || [[Base_6|6]] || 3,4
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| |-----
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| | 2 || 6 || 13,44 || 9, 28
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| |-----
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| | 4 || 6 || 0213,5344 || 81, 1216
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| |-----
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| | 1 || [[Base_10|10]] || 5,6
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| |-----
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| | 2 || 10 || 25,76
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| |-----
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| | 4 || 10 || 0625,9376
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| |-----
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| | 1 || [[Base_12|12]] || 4,9
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| |-----
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| | 2 || 12 || 54,69 || 64, 81
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| |-----
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| | 3 || 12 || 854,369 || 1216, 513
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| |-----
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| | 4 || 12 || 3854,8369 || 6400, 14337
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| |-----
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| | 1 || [[Base_14|14]] || 7,8
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| |-----
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| | 2 || 14 || 37,A8 || 49, 148
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| |-----
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| | 4 || 14 || 0C37,D1A8 || 2401, 344
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| |-----
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| | 1 || [[Base_15|15]] || 6,A || 6,10
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| |-----
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| | 2 || 15 || 86,6A || 126, 100
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| |-----
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| | 1 || [[Base_18|18]] || 9,A || 9, 10
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| |-----
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| | 2 || 18 || 49,DA || 81, 244
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| |-----
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| | 4 || 18 || 1249,GFDA || 6561, 98416
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| |-----
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| | 1 || [[Base_24|24]] || 9,G || 9, 16
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| |-----
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| | 2 || 24 || M9,2G || 513, 64
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| |-----
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| | 4 || 24 || D0M9,AP2G || 180225, 151552
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| |-----
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| | 1 || 30 || 6,A,F,G,M,R || 6, 10, 15, 16, 21, 25
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| |-----
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| | 2 || 30 || K6,3A,7F,NG,SM,AR || 576, 100, 225, 676, 801, 325
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| |-----
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| | 4 || 30 || B2K6,H13A,1S7F,U3NG,CUSM,JTAR || 299376, 460000, 50625, 759376, 350001, 510625
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| |}
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| Note that the base 30 numbers expressed in decimal are also automorphic in the last 4 digits.
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| == External links ==
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| *{{MathWorld | urlname=AutomorphicNumber | title=Automorphic number}}
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| == References ==
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| http://planetmath.org/examplesof1automorphicnumbers | |
| {{Classes of natural numbers}}
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| [[Category:Base-dependent integer sequences]]
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